context

Current status of research according to AI

/ Paul Pukite (@whut) / 3 Comments

explain the Pukite tidal theory

The Pukite Tidal Theory, primarily developed by researcher Paul Pukite, proposes that long-period tidal forcing is the underlying driver for several major geophysical and atmospheric cycles that have previously been considered erratic or unresolved. [1, 2]

The core of the theory is that small gravitational perturbations from the Moon and Sun, which are perfectly predictable, are “aliased” or modulated by seasonal cycles to create the complex behaviors seen in Earth’s systems. [3, 4]

Key Phenomena Addressed

Pukite applies this model to three main “unresolved mysteries” in geophysics:

  • Quasi-Biennial Oscillation (QBO): A regular reversal of stratospheric winds. The theory argues that lunar nodal cycles, when combined with the annual solar cycle, create the roughly 28-month QBO period through a process called physical aliasing.
  • El Niño Southern Oscillation (ENSO): An erratic oceanic temperature cycle. The model suggests ENSO is a “sloshing” response of the ocean to tractive gravitational forces, essentially treating it as a solution to Laplace’s Tidal Equations.
  • Chandler Wobble: A small deviation in the Earth’s axis of rotation. The theory posits this is caused by an external lunar torque rather than internal Earth dynamics. [1, 2, 3, 5, 6, 7]

How the Model Works

  1. Lunar Gravitational Potential: The model starts by calculating the precise lunar gravitational potential over time.
  2. Seasonal Modulation: These fast lunar cycles are provoked by a seasonal (yearly) peak in energy. This “carrier” signal helps filter out short-term fluctuations and reveals long-term patterns.
  3. Physical Aliasing: Because these cycles are sampled or triggered by seasonal events (like solstices), the resulting data shows lower-frequency “aliased” harmonics that match the observed multi-year periods of ENSO and QBO. [4, 8, 9]

Scientific Reception

This theory is considered a novel and controversial alternative to standard geophysical models. While it offers high correlation with historical data, it has faced skepticism from mainstream physical oceanographers and meteorologists who argue that it may “shoehorn” complex fluid dynamics into a periodic framework. Much of the work is documented on the GeoEnergy Math blog and in discussion papers for the Earth System Dynamics (ESD) journal. [1, 4, 7, 10]

Would you like to see a more detailed breakdown of the mathematical aliasing used to explain the QBO’s 28-month cycle?

[1] https://esd.copernicus.org

[2] https://www.researchgate.net

[3] https://esd.copernicus.org

[4] https://geoenergymath.com

[5] https://acp.copernicus.org

[6] https://geoenergymath.com

[7] https://geoenergymath.com

[8] https://esd.copernicus.org

[9] https://geoenergymath.com

[10] https://esd.copernicus.org

This link may go stale, but here is the original response:

https://share.google/aimode/ta098ixUIyoNefp49

The perplexing Chandler wobble

/ Paul Pukite (@whut) / 1 Comment

I had heard that Perplexity was an LLM geared to scientific research, so I gave it this prompt:

Explain the Chandler wobble as due to a stroboscopic driving force emerging from the nonlinear modulation of an impulsed annual signal with the periodic declination torque of the lunar draconic cycle operating as a wavenumber=0 group symmetry.

The Chandler wobble can be viewed as a stroboscopically forced precession mode of Earth’s rotation, driven by a nonlinear interaction between an annually pulsed excitation and the lunar draconic (nodal) tide, with the effective forcing emerging in the band of the 433‑day free wobble. [1,2,3]

The behavioral characteristics include a cycle and damping:

  • The Chandler wobble is a free nutation of Earth’s rotation axis with a period of about 433 days and amplitude of order 0.05–0.2 arcsec, slightly different from the 305‑day Euler period because Earth is elastic and has oceans and a fluid core. [3,6,7]
  • In the absence of forcing, this mode would decay on a timescale of decades, so it must be continuously re‑excited by some geophysical process (mass redistribution in oceans/atmosphere, etc.). [3,4,5]

An annual impulsive forcing also generates an annual wobble. Think of the annual cycle as supplying near‑impulsive changes in the effective torque that can couple into polar motion, because seasonal mass shifts and circulation changes are strongest at certain times of year (e.g., around solstices). [1,3,5]

In a simplified model, this is approximated as an annual train of impulses in the excitation function of the polar motion equation rather than a smooth sinusoid. [1,2]

An impulse train has a flat, comb‑like spectrum: it projects power into many sidebands around any underlying carrier frequency it modulates. When that carrier is the lunar draconic tide, the sidebands can “step” into the Chandler band by aliasing and nonlinear mixing. [1,2]

The draconic (nodal) tide acts as a modulator to force sub-band cycles on the annual carrier. The lunar draconic cycle (node–to–node motion of the Moon’s orbit relative to the equatorial plane) produces a strong, periodic variation in declination and thus in the latitude pattern of tidal torque on Earth.  The relevant subharmonic here is the draconic fortnight (≈13.606 days), which governs how the lunar tidal bulge migrates north–south. [1]

If you take that relatively high frequency draconic signal and modulate it with a biannual or annual envelope tied to the solar geometry (solstice alignment of Sun–Moon–Earth), you get an amplitude‑modulated carrier. In Fourier terms, that modulation splits the draconic line into a comb of spectral lines spaced by ± n × (1 yr) around the original frequency. [1,2]

Because Earth’s free Chandler frequency sits far below the raw draconic frequency, you don’t resonate with the carrier itself; instead you hit the Chandler band through an aliasing condition: one of those sidebands (carrier ± k×annual) falls extremely close to the Chandler frequency (≈0.843 cycles/yr), so energy is pumped efficiently into the wobble. [1,2]

There’s a stroboscopic viewpoint to aid in understanding. From a stroboscope’s perspective, the annual (or twice‑annual) sampling by the Sun–Earth geometry “views” the draconic torque at almost, but not exactly, the same phase each year.  This is analogous to watching a fast‑spinning wheel under a flashing light: if the flash frequency is close but not equal to the spin frequency, you see a slow apparent beat motion. The slow beat corresponds here to the Chandler period, and the flashing is the annually pulsed excitation. [1,2]

Mathematically, the product of a fast oscillation (draconic tide) and an impulsive annual envelope produces cross‑terms whose effective frequency is the difference between the draconic and integer multiples of the annual frequency. One of these difference frequencies lands in the Chandler band, so in the rotating‑frame equations for polar motion it appears as a nearly resonant, narrowband forcing. [1,2,3]

Concerning the Wavenumber = 0 group symmetry: Polar motion is effectively a global (degree‑1, order‑1 in spherical harmonics) displacement of Earth’s rotation axis with respect to its figure; the Chandler mode is a normal mode of the whole body. [5,6,7]

The particular draconic–annual interaction you describe can be interpreted as exciting the azimuthally symmetric part of the torque pattern—no preferred longitude, just a latitude‑dependent load—which in spectral language is a wavenumber‑0 (axisymmetric) component in Earth’s rotating frame. [1,6]

Calling it a “wavenumber=0 group symmetry” is a way of saying: after averaging over the fast draconic and diurnal phases, what survives is the group‑velocity‑scale envelope of the forcing, which is spatially uniform in longitude and thus couples efficiently to the global Chandler eigenmode while filtering out higher‑wavenumber patterns that would average to zero. [1,6,7]

So in compact form: the Chandler wobble emerges as a resonant response of Earth’s free polar mode to a stroboscopically generated, axisymmetric forcing produced by nonlinear mixing of an annually impulsed excitation with the lunar draconic tide, whose aliased sideband falls at the Chandler frequency. [1,2,3]

(continuation prompt)

What mathematical formula derives the 433-day period from draconic aliasing?

You can get the 433‑day Chandler period as a beat (alias) period between the draconic fortnightly tide and the annual cycle by using the standard aliasing / beat‑frequency relation in frequency space. [1,2,3]

1. Set up the frequencies.  Let

Draconic fortnight period (half draconic month)
T_{d} = 27.2122/2 \approx 13.6061 days. [1,2]

Annual period
T_{y} = 365.242] days. [1,4]

Convert to frequencies (in cycles per day):

f_d = \frac{1}{T_d}, f_y = \frac{1}{T_y}

Numerically, [1,2]

f_d = \frac{1}{13.60611} = 0.0735 c/d, f_y = \frac{1}{365.2421} = 0.0027379 c/d.

2. Alias (beat) with annual sampling is when you modulate or impulsively sample the draconic signal annually, sidebands appear at

f = |f_d - n f_y|, n = 1, 2, 3, . . .

The effective long period seen in the slowly varying envelope (the alias period) is

T_\text{alias} = \frac{1}{|f_d - n f_y|}

Choose integer [n] such that T_{\text{alias}} lands near 433 days. In the GeoEnergy Math formulation this corresponds to the harmonic that comes out closest to the Chandler band, giving

T_\text{alias} = 432.77 \text{days} = 1.185 \text{years},

which matches the observed Chandler wobble period. [1,2,3]

So, in words: the mathematical formula is the beat‑period relation

T_{\text{alias}} = 1 / |f_{d} - n f_{y}|,

with f_{d} from the 13.606‑day draconic fortnight and f_{y} from the annual cycle, and for the appropriate harmonic (n) this evaluates to ≈ 433 days. [1,2,3]

Refs

  1. https://geoenergymath.com/2017/03/04/the-chandler-wobble-challenge/
  2. https://geoenergymath.com/2017/05/21/the-lunar-geophysical-connection/
  3. https://esd.copernicus.org/preprints/esd-2020-74/esd-2020-74-AC5.pdf
  4. https://en.wikipedia.org/wiki/Chandler_wobble
  5. http://landmarks.allenbrowne.info/LatitudeObservatory/ChandlerWobble.htm
  6. https://legacy.iho.int/iho_pubs/CB/C-33/C-33_maree_simon_en.pdf
  7. https://www.technologyreview.com/2009/08/31/265599/earths-chandler-wobble-changed-dramatically-in-2005/

So, it basically summarized easily available sources from the internet, mainly from what I had written before.

Hidden latent manifolds in fluid dynamics

/ Paul Pukite (@whut) / 3 Comments

The behavior of complex systems, particularly in fluid dynamics, is traditionally described by high-dimensional systems of equations like the Navier-Stokes equations. While providing practical applications as is, these models can obscure the underlying, simplified mechanisms at play. It is notable that ocean modeling already incorporates dimensionality reduction built in, such as through Laplace’s Tidal Equations (LTE), which is a reduced-order formulation of the Navier-Stokes equations. Furthermore, the topological containment of phenomena like ENSO and QBO within the equatorial toroid , and the ability to further reduce LTE in this confined topology as described in the context of our text Mathematical Geoenergy underscore the inherent low-dimensional nature of dominant geophysical processes. The concept of hidden latent manifolds posits that the true, observed dynamics of a system do not occupy the entire high-dimensional phase space, but rather evolve on a much lower-dimensional geometric structure—a manifold layer—where the system’s effective degrees of freedom reside. This may also help explain the seeming paradox of the inverse energy cascade, whereby order in fluid structures seems to maintain as the waves become progressively larger, as nonlinear interactions accumulate energy transferring from smaller scales.

Discovering these latent structures from noisy, observational data is the central challenge in state-of-the-art fluid dynamics. Enter the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, pioneered by Brunton et al. . SINDy is an equation-discovery framework designed to identify a sparse set of nonlinear terms that describe the evolution of the system on this low-dimensional manifold. Instead of testing all possible combinations of basis functions, SINDy uses a penalized regression technique (like LASSO) to enforce sparsity, effectively winnowing down the possibilities to find the most parsimonious, yet physically meaningful, governing differential equations. The result is a simple, interpretable model that captures the essential physics—the fingerprint of the latent manifold. The SINDy concept is not that difficult an algorithm to apply as a decent Python library is available for use, and I have evaluated it as described here.

Applying this methodology to Earth system dynamics, particularly the seemingly noisy, erratic, and perhaps chaotic time series of sea-level variation and climate index variability, reveals profound simplicity beneath the complexity. The high-dimensional output of climate models or raw observations can be projected onto a model framework driven by remarkably few physical processes. Specifically, as shown in analysis targeting the structure of these time series, the dynamics can be cross-validated by the interaction of two fundamental drivers: a forced gravitational tide and an annual impulse.

The presence of the forced gravitational tide accounts for the regular, high-frequency, and predictable components of the dynamics. The annual impulse, meanwhile, serves as the seasonal forcing function, representing the integrated effect of large-scale thermal and atmospheric cycles that reset annually. The success of this sparse, two-component model—where the interaction of these two elements is sufficient to capture the observed dynamics—serves as the ultimate validation of the latent manifold concept. The gravitational tides with the integrated annual impulse are the discovered, low-dimensional degrees of freedom, and the ability of their coupled solution to successfully cross-validate to the observed, high-fidelity dynamics confirms that the complex, high-dimensional reality of sea-level and climate variability emerges from this simple, sparse, and interpretable set of latent governing principles. This provides a powerful, physics-constrained approach to prediction and understanding, moving beyond descriptive models toward true dynamical discovery.

An entire set of cross-validated models is available for evluation here: https://pukpr.github.io/examples/mlr/.

This is a mix of climate indices (the 1st 20) and numbered coastal sea-level stations obtained from https://psmsl.org/

https://pukpr.github.io/examples/map_index.html

  • nino34 — NINO34 (PACIFIC)
  • nino4 — NINO4 (PACIFIC)
  • amo — AMO (ATLANTIC)
  • ao — AO (ARCTIC)
  • denison — Ft Denison (PACIFIC)
  • iod — IOD (INDIAN)
  • iodw — IOD West (INDIAN)
  • iode — IOD East (INDIAN)
  • nao — NAO (ATLANTIC)
  • tna — TNA Tropical N. Atlantic (ATLANTIC)
  • tsa — TSA Tropical S. Atlantic (ATLANTIC)
  • qbo30 — QBO 30 Equatorial (WORLD)
  • darwin — Darwin SOI (PACIFIC)
  • emi — EMI ENSO Modoki Index (PACIFIC)
  • ic3tsfc — ic3tsfc (Reconstruction) (PACIFIC)
  • m6 — M6, Atlantic Nino (ATLANTIC)
  • m4 — M4, N. Pacific Gyre Oscillation (PACIFIC)
  • pdo — PDO (PACIFIC)
  • nino3 — NINO3 (PACIFIC)
  • nino12 — NINO12 (PACIFIC)
  • 1 — BREST (FRANCE)
  • 10 — SAN FRANCISCO (UNITED STATES)
  • 11 — WARNEMUNDE 2 (GERMANY)
  • 14 — HELSINKI (FINLAND)
  • 41 — POTI (GEORGIA)
  • 65 — SYDNEY, FORT DENISON (AUSTRALIA)
  • 76 — AARHUS (DENMARK)
  • 78 — STOCKHOLM (SWEDEN)
  • 111 — FREMANTLE (AUSTRALIA)
  • 127 — SEATTLE (UNITED STATES)
  • 155 — HONOLULU (UNITED STATES)
  • 161 — GALVESTON II, PIER 21, TX (UNITED STATES)
  • 163 — BALBOA (PANAMA)
  • 183 — PORTLAND (MAINE) (UNITED STATES)
  • 196 — SYDNEY, FORT DENISON 2 (AUSTRALIA)
  • 202 — NEWLYN (UNITED KINGDOM)
  • 225 — KETCHIKAN (UNITED STATES)
  • 229 — KEMI (FINLAND)
  • 234 — CHARLESTON I (UNITED STATES)
  • 245 — LOS ANGELES (UNITED STATES)
  • 246 — PENSACOLA (UNITED STATES)

Crucially, this analysis does not use the SINDy algorithm, but a much more basic multiple linear regression (MLR) algorithm predecessor, which I anticipate being adapted to SINDy as the model is further refined. Part of the rationale for doing this is to maintain a deep understanding of the mathematics, as well as providing cross-checking and thus avoiding the perils of over-fitting, which is the bane of neural network models.

Also read this intro level on tidal modeling, which may form the fundamental foundation for the latent manifold: https://pukpr.github.io/examples/warne_intro.html. The coastal station at Wardemunde in Germany along the Baltic sea provided a long unbroken interval of sea-level readings which was used to calibrate the hidden latent manifold that in turn served as a starting point for all the other models. Not every model works as well as the majority — see Pensacola for a sea-level site and and IOD or TNA for climate indices, but these are equally valuable for understanding limitations (and providing a sanity check against an accidental degeneracy in the model fitting process) . The use of SINDy in the future will provide additional functionality such as regularization that will find an optimal common-mode latent layer,.

Simpler models can outperform deep learning at climate prediction

/ Paul Pukite (@whut) / 3 Comments

This article in MIT News:

https://news.mit.edu/2025/simpler-models-can-outperform-deep-learning-climate-prediction-0826

“New research shows the natural variability in climate data can cause AI models to struggle at predicting local temperature and rainfall.” … “While deep learning has become increasingly popular for emulation, few studies have explored whether these models perform better than tried-and-true approaches. The MIT researchers performed such a study. They compared a traditional technique called linear pattern scaling (LPS) with a deep-learning model using a common benchmark dataset for evaluating climate emulators. Their results showed that LPS outperformed deep-learning models on predicting nearly all parameters they tested, including temperature and precipitation.“

Machine learning and other AI approaches such as symbolic regression will figure out that natural climate variability can be done using multiple linear regression (MLR) with cross-validation (CV), which is an outgrowth or extension of linear pattern scaling (LPS).

https://pukpr.github.io/results/image_results.html

“When this was initially created on 9/1/2025, there were 3000 CV results on time-series
that averaged around 100 years (~1200 monthly readings/set) so over 3 million data points“

In this NINO34 (ENSO) model, the test CV interval is shown as a dashed region

I developed this github model repository to make it easy to compare many different data sets, much better than using an image repository such as ImageShack.

There are about 130 sea-level height monitoring stations in the sites, which is relevant considering how much natural climate variation a la ENSO has an impact on monthly mean SLH measurements. See this paper Observing ENSO-modulated tides from space

“In this paper, we successfully quantify the influences of ENSO on tides from multi-satellite altimeters through a revised harmonic analysis (RHA) model which directly builds ENSO forcing into the basic functions of CHA. To eliminate mathematical artifacts caused by over-fitting, Lasso regularization is applied in the RHA model to replace widely-used ordinary least squares. “

Model Ontology

/ Paul Pukite (@whut) / 2 Comments

In Chapter 10 of the book we touch on organization of environmental models.

“Furthermore, by applying ontology‐based approaches for organizing models and techniques, we can set the stage for broader collections of such models discoverable by a general community of designers and analysts. Together with standard access protocols for context modeling,
these innovations provide the promise of making environmental context models generally available and reusable, significantly assisting the energy analyst.”

Energy Transition : Applying Probabilities and Physics

Although we didn’t elaborate on this topic, it is an open area for future development, as our 2017 AGU presentation advocates. The complete research report is available as https://doi.org/10.13140/RG.2.1.4956.3604.

What we missed on the first pass was an ontology for citations titled CiTO (Citation Typing Ontology) which enables better classification and keeping track of research lineage. The idea again is to organize and maintain scientific knowledge for engineering and scientific modeling applications. As an example, one can readily see how the Citation Typing Ontology could be applied, with the is_extended_by object property representing much of how science and technology advances — in other words, one finding leading to another.

The ENSO Forcing Potential – Cheaper, Faster, and Better

/ Paul Pukite (@whut) / Leave a comment

Following up on the last post on the ENSO forcing, this note elaborates on the math.  The tidal gravitational forcing function used follows an inverse power-law dependence, where a(t) is the anomalistic lunar distance and d(t) is the draconic or nodal perturbation to the distance.

F(t) propto frac{1}{(R_0 + a(t) + d(t))^2}'

Note the prime indicating that the forcing applied is the derivative of the conventional inverse squared Newtonian attraction. This generates an inverse cubic formulation corresponding to the consensus analysis describing a differential tidal force:

F(t) propto -frac{a'(t)+d'(t)}{(R_0 + a(t) + d(t))^3}

For a combination of monthly and fortnightly sinusoidal terms for a(t) and d(t) (suitably modified for nonlinear nodal and perigean corrections due to the synodic/tropical cycle)   the search routine rapidly converges to an optimal ENSO fit.  It does this more quickly than the harmonic analysis, which requires at least double the unknowns for the additional higher-order factors needed to capture the tidally forced response waveform. One of the keys is to collect the chain rule terms a'(t) and d'(t) in the numerator; without these, the necessary mixed terms which multiply the anomalistic and draconic signals do not emerge strongly.

As before, a strictly biennial modulation needs to be applied to this forcing to capture the measured ENSO dynamics — this is a period-doubling pattern observed in hydrodynamic systems with a strong fundamental (in this case annual) and is climatologically explained by a persistent year-to-year regenerative feedback in the SLP and SST anomalies.

Here is the model fit for training from 1880-1980, with the extrapolated test region post-1980 showing a good correlation.

The geophysics is now canonically formulated, providing (1) a simpler and more concise expression, leading to (2) a more efficient computational solution, (3) less possibility of over-fitting, and (4) ultimately generating a much better correlation. Alternatively, stated in modeling terms, the resultant information metric is improved by reducing the complexity and improving the correlation — the vaunted  cheaper, faster, and better solution. Or, in other words: get the physics right, and all else follows.

 

 

 

 

 

 

 

 

 

 

 

 

 

Solar Eclipse 2017 : What else?

/ Paul Pukite (@whut) / 1 Comment

The reason we can so accurately predict the solar eclipse of 2017 is because we have accurate knowledge of the moon’s orbit around the earth and the earth’s orbit around the sun.

Likewise, the reason that we could potentially understand the behavior of the El Nino Southern Oscillation (ENSO) is that we have knowledge of these same orbits. As we have shown and will report at this year’s American Geophysical Union (AGU) meeting, the cyclic gravitational pull of the moon (lower panel in Figure 1 below) interacting seasonally precisely controls the ENSO cycles (upper panel Figure 1).

Fig 1: Training interval 1880-1950 leads to extrapolated fit post-1950

Figure 2 is how sensitive the fit is to the precise value of the lunar cycle periods. Compare the best ft values to the known lunar values here. This is an example of the science of metrology.

Fig 2: Sensitivity to selection of lunar periods.

The implications of this research are far-ranging. Like knowing when a solar eclipse occurs helps engineers and scientists prepare power utilities and controlled climate experiments for the event, the same considerations apply to ENSO.  Every future El Nino-induced heat-wave or monsoon could conceivably be predicted in advance, giving nations and organizations time to prepare for accompanying droughts, flooding, and temperature extremes.

Follow @whut on Twitter:

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ENSO Split Training for Cross-Validation

/ Paul Pukite (@whut) / 10 Comments

If we split the modern ENSO data into two training intervals — one from 1880 to 1950 and one from 1950 to 2016, we get roughly equal-length time series for model evaluation.

As Figure 1 shows, a forcing stimulus due to monthly-range LOD variations calibrated to the interval between 2000 to 2003 (lower panel) is used to train the ENSO model in the interval from 1880 to 1950. The extrapolated model fit in RED does a good job in capturing the ENSO data in the period beyond 1950.

Fig. 1: Training 1880 to 1950

Next, we reverse the training and verification fit, using the period from 1950 to 2016 as the training interval and then back extrapolating. Figure 2 shows this works about as well.

Fig. 2: Training interval 1950 to 2016

Continue reading →

Deterministic and Stochastic Applied Physics

/ Paul Pukite (@whut) / Leave a comment

Pierre-Simon Laplace was one of the first mathematicians who took an interest in problems of probability and determinism.  It’s surprising how much of the math and applied physics that Laplace developed gets used in day-to-day analysis. For example, while working on the ENSO and QBO analysis, I have invoked the following topics at some point:

  1. Laplace’s tidal equations
  2. Laplace’s equation
  3. Laplacian differential operator
  4. Laplace transform
  5. Difference equation
  6. Planetary and lunar orbital perturbations
  7. Probability methods and problems
    1. Inductive probability
    2. Bayesian analysis, e.g. the Sunrise problem
  8. Statistical methods and applications
    1. Central limit theorem
    2. Least squares
  9. Filling in holes of Newton’s differential calculus
  10. Others here

Apparently he did so much and was so comprehensive that in some of his longer treatises he often didn’t cite the work of others, making it difficult to pin down everything he was responsible for (evidently he did have character flaws).

In any case, I recall applying each of the above in working out some aspect of a problem. Missing was that Laplace didn’t invent Fourier analysis but the Laplace transform is close in approach and utility.

When Laplace did all this research, he must have possessed insight into what constituted deterministic processes:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

— Pierre Simon Laplace,
A Philosophical Essay on Probabilities[wikipedia]
This is summed up as:

He also seemed to be a very applied mathematician, as per a quote I have used before  “Probability theory is nothing but common sense reduced to calculation.”  Really nothing the least bit esoteric about any of Laplace’s math, as it seemed always motivated by solving some physics problem or scientific observation. It appears that he wanted to explain all these astronomic and tidal problems in as simple a form as possible. Back then it may have been esoteric, but not today as his techniques have become part of the essential engineering toolbox. I have to wonder if Laplace were alive now whether he would agree that geophysical processes such as ENSO and QBO were equally as deterministic as the sun rising every morning or of the steady cyclic nature of the planetary and lunar orbits. And it wasn’t as if Laplace possessed confirmation bias that behaviors were immediately deterministic; as otherwise he wouldn’t have spent so much effort in devising the rules of probability and statistics that are still in use today, such as the central limit theorem and least squares.

Perhaps he would have glanced at the ENSO problem for a few moments, noticed that in no way that it was random, and then casually remarked with one his frequent idiomatic phrases:

“Il est aisé à voir que…”  … or ..  (“It is easy to see that…”).

It may have been so obvious that it wasn’t important to give the details at the moment, only to fill in the chain of reasoning later.  Much like the contextEarth model for QBO, deriving from Laplace’s tidal equations.

Where are the Laplace’s of today that are willing to push the basic math and physics of climate variability as far as it will take them? It has seemingly jumped from Laplace to Lorenz and then to chaotic uncertainty ala Tsonis or mystifying complexity ala Lindzen. Probably can do much better than to punt like that … on first down even !

Scaling El Nino

/ Paul Pukite (@whut) / 3 Comments

Recently, the rock climber Alex Honnold took a route up El Capitan without ropes.There’s no room to fail at that. I prefer a challenge that one can fail at, and then keep trying.  This is the ascent to conquering El Nino:

The Free-thought Route*

Χ  Base camp:  ENSO (El Nino/Southern Oscillation) is a sloshing behavior, mainly in the thermocline where the effective gravity makes it sensitive to angular momentum changes.
Χ  Faster forcing cycles reinforce against the yearly cycle, creating aliased periods. How?
Χ  Monthly lunar tidal cycles provide the aliased factors: Numbers match up perfectly.
① This aliasing also works for QBO, an atmospheric analog of ENSO.
Χ  A biennial meta-stability appears to be active. Cycles reinforce on alternating years.
Χ  The well-known Mathieu modulation used for sloshing simulations also shows a biennial character.
② Machine learning experiments help ferret out these patterns.
Χ  The delay differential equation formulation matches up with the biennial Mathieu modulation with a delay of one-year.  That’s the intuitive yearly see-saw that is often suggested to occur.
③  The Chandler wobble also shows a tidal forcing tendency, as does clearly the earth’s LOD (length-of-day) variations.
Χ  Integrating the DiffEq model provides a good fit, including long-term coral proxy records
Χ  Solving the Laplace tidal equation via a Sturm-Liouville expression along the equator helps explain details of QBO and ENSO
④  Close inspection of sea-level height (SLH) tidal records show evidence of both biennial and ENSO characteristics
Δ Summit: Final validation of the geophysics comparing ENSO forcing against LOD forcing.

Model fits to ENSO using a training interval

The route encountered several dead-ends with no toe-holds or hand-holds along the way (e.g. the slippery biennial phase reversal, the early attempts at applying Mathieu equation). In retrospect many of these excursions were misguided or overly complex, but eventually other observations pointed to the obvious route.

This is a magnification of the fitting contour around the best forcing period values for ENSO. These pair of peak values are each found to be less than a minute apart from the known values of the Draconic cycle (27.2122 days) and Anomalistic cycle (27.5545 days).

The forcing comes directly from the angular momentum variations in the Earth’s rotation. The comparison between what the ENSO model uses and what is measured via monitoring the length-of-day (LOD) is shown below:

 

 

*  This is not the precise route I took, but how I wish it was in hindsight.